EOM for Structures in Terms of Quasi Coordinates

8/4/2019 EOM for Structures in Terms of Quasi Coordinates

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EQUATIONS OF MOTION FOR STRUCTURESIN TERMS OF QUASICOORDINATES

Appears in theJournal of Applied HechanicsVol. 57, No. 3, pp.745-749September 1990

Roger D. QuinnGeneral Motors Assistant ProfessorMechanical And Aerospace EngineeringCase Western Reserve UniversityCleveland. Ohio 44106

.-CT

A form of Lagrange's Equations in terms of quasicoordinates(Boltzmann/Hamel equations) is presented. Identities are introducedwhich permit a straightforward formulation of the equations of motionfor structures for which the kinetic and potential energies arefunctions of angular velocity and orientation. The formalism ispresented in matrix form and may be used if the energies are expressedin matrix form as explicit functions of angular velocities andcoordinate transformation matrices. This method is particularly usefulfor a large class of problems in the dynamics of structures includingspacecraft, robots, ground vehicles and aircraft.


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1. IntroductionThe purpose of this paper is to present a form of tagrange's

equations in terms of quasicoordinates (BoltzmandHamel equations).This form is specific to a large class of problems in the dynamics ofstructures including spacecraft, robots, ground vehicles and aircraftwhere the angular velocities of the structure can be considered to betime derivatives of quasicoordinates. Identities are introduced whichpermit a straightforward method of formulating the equations of motionwhen the kinetic and potential energies are explicit functions ofangular orientation. The formulation is presented in matrix form sothat the kinetic and potential energies need to be expressed in matrixform as functions of angular velocities and coordinate transformationmatrices.

The concept of quasi-coordinates is not new. According toWhittaker (1944) and Neimark and Fufaev (19671, Lagrange and Euler usedquasicoordinates to study rigid body motion and the so-called Lagrange'sequations for quasicoordinates were developed by Boltzmann and Hamel atthe beginning of the twentieth century. Hence, this Lagrangianformalism is sometimes called the BoltzmannAIamel equations. Advancedtexts in dynamics such as those by Whittaker (1944) and Meirovitch(1970) include a section on the subject; and quasicoordinateformulations have been recently used by Passeron et. al. (1986). Huston

and Passerello (19801, and Oz et. al. (1980). However, the treatment indynamics texts is usually of a general nature and, as such, asimplification which enhances the utility of the method does not appearin the literature.

Lagrange's equations are widely used because they provide (i) a


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straightforward and orderly analytical approach based on a single scalarfunction which produces (ii) form-invariant equations of motion in which(iii) holonomic constraint forces can be eliminated, included orretrieved, as desired, and (iv) natural symmetry is preserved.According to Neimark and Fufaev (19671, Lagrange' equations forquasicoordinates were developed as a form which was uniformly valid forall dynamic systems with our without constraints for true orquasi-coordinates. In practical terms, Lagrange's equations forquasicoordinates permit a most efficient formulation of the equations ofmotion of some systems. Perhaps, the clearest (and earliest) example isthat of a structure undergoing finite rotations in three-dimensionalspace. If true generalized coordinates are chosen to represent theserotations, the derivation of the equations of motion using Lagrange'sequations becomes quite tedious. In the quasicoordinate formulation,the angular velocity vector components are the time derivatives of a setof quasicoordinates rather than true coordinates; thesequasicoordinates are defined only in terms of their differentials. Thisapproach is based on the observation that the kinetic energy of a bodycan be represented in its most compact form in terms of these angularvelocity components or quasivelocities.

It is important to note that the validity of the equations ofmotion of a dynamic system depends on the system model used in theirdevelopment, not on the particular method of formulation, as there aremany theoretically equivalent methods (all based on Lagrange's principleof virtual work) as discussed by Likins (1974, 1975). However, the formof the resulting equations may be different and considerablemathematical manipulations may be necessary for comparisons. The best


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method of equation of motion development for a particular dynamic systemproduces the simplest and most useful (e.g. symmetry preserved) form ofthe equations in an orderly fashion. It is for this reason offormulation efficiency that the Boltzmann/Hamel equation has beenexamined.

General forms of Lagrange's equations in quasicoordinate formappear ungainly. It is when special cases are examined that theseequations appear most promising. An important special case is thatmentioned earlier where a structure undergoes finite rotations and thekinetic and potential energies are not functions of the angularorientation. The purpose of this paper is to present simplifyingidentities which enhance the usefulness of this approach in cases wherethe energies are explicit functions of the angular orientation.

In the kinematic analysis of a structure, it is often convenient toexpress different velocity and position vectors in different referenceframes rather than all in the same frame. Hence, when the energies areexpressed in matrix form, coordinate transformation matrices must beintroduced. These coordinate transformation matrices and, hence, theenergies are functions of the relative angular orientation of thereference frames. For this reason, a quasicoordinate form of Lagrange'sequations which can easily account for terms involving angularorientation is valuable.

This formulation of Lagrange's equations for quasicoordinates isclearly useful for the dynamic analysis of robots, spacecraft, groundvehicles and aircraft. The equations governing the rotational motion ofa maneuvering flexible spacecraft are developed using this approach.The purpose of this example is to demonstrate this method of dynamic


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analysis for a problem of current interest, the solution of which hasbeen published previously, for comparison.

2. General Form of Lagrange's Equations For QuasicoordinatesLagrange's equations of motion for holonomic systems in terms of

true generalized coordinates can be expressed in the matrix form

where q is a set of n true independent generalized coordinates, T and V-are the kinetic and potential energies where T= T ( ~ , ~ ) .=V(q) and Q is a- - -set of generalized forces which are defined in terms of the virtual workexpression

1su = q sg (2Lagrange's equations in the form of Eq. (1) provide a straightforwardmethod of deriving reactionless equations of motion for dynamic systems.However, when a structure is free to undergo finite rotations,derivation of the equations governing the rigid-body orientation in thismanner can become tedious. For this reason other forms of Lagrange'sequations have been developed.

The most compact form of the kinetic energy of a structureundergoing finite rotations is expressed in terms of the inertialangular velocity vector. Hence, we night conclude that the most

efficient derivation of the equations of motion must also be in terms ofthese angular velocities. For this reason a concept was developed wherethe angular velocity vector is considered to be the time derivatives ofso-called quasicoordinates which are of themselves undefined.

That part of Eq. (1) which governs the orientation of a structure


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can be expressed as

where a is a set of three true coordinates such as Euler angles and !a"is a set of corresponding moments. The angular velocity vector can beexpressed as a function of the true coordinates and their timederivatives or in matrix form

where D(a) is a 3x3 matrix which is a nonlinear function of a. The- -kinetic energy can then be expressed as an explicit function of w and a.- -or T(w,a). The introduction of the inverse of Eq. ( 4 ) into Eq. (3)- a,permits Lagrange's equations to be represented in terms of w. In thismanner Lagrange's equations of motion for rigid body rotations can beexpressed as

where the following matrix notation has been used:

and D - ~ = DT1-'.Premultiplying Eq. 15) by 0-'. Lagrange's equations for

quasicoordinates can be expressed as

This form of Lagrange's equations appears in Whittaker (1944) (inindicia1 notation), Meirovitch (1970) and Likins (1975). Although, inthe development of Eq. (71, rotating structures have been discussed, theresulting equations are of general utility in that the symbols a and w


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could take other meanings. Equation ( 7 ) is actually more complex thanEq. (3) and, in practice, can be rather cumbersome to apply.Fortunately, for the class of problems considered in this paper Eq. (7)can be simplified greatly.

3. Quasicoordinate Formulations for Rotating StructuresIn the case of structures undergoing finite rotations in three

dimensional space, the angular velocities can be further related to thematrix D to simplify Eq. (7).

Let the matrix C(a) denote the orthogonal rotational transformation-of a structure from the inertial reference frame to a body-fixedreference frame or v = C v where v is an arbitrary vector and the- -N -subscripts B and N represent the body-fixed and inertial frames,respectively.. The time rate of change of this matrix can be expressed

where w is a skew symmetric matrix defined as

Solving Eq. (8a) for 6, e have

where

The introduction of Eq. (4 ) into Eq. (gal produces an explicitrelationship between the matrices C and D. This permits the proof ofthe following identity which is implicit in Meirovitch (1970):


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Introducing Eq. (10) into Eq. (71, Lagrange's equations forquasi-coordinates take the form

where

is the moment which can be defined in terms of virtual work as

where /3 is the set of quasicoordinates which are defined in differential-form as w = d#3/dt. In the literature, attention is focused on cases" -where T and V are not functions of the orientation a so that the last"term on the left side of Eq. (11) is null. In this case, the result isthe familiar set of equations that are often derived from angularmomentum principles. On the other hand, if the last term is not null,the utilization of Eq. (11) remains rather cumbersome.

In general, the potential and kinetic energies are functions of theangular orientation of the structure. In matrix form, ?: and V containrotational transformation matrices which depend explicitly on angularorientation. A term of the kinetic energy might take the matrix form

where b and n are vectors represented in the body-fixed and inertial- "reference frames, respectively. In this case, the last term on the leftside of Eq. (1 11 involving ~ ? w . a )- can be expressed as

where the notation of Eq. ( 6 ) has been used to define the terms on the


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right hand side of Eq. (14). Moreover, it can be shown that

*T TTo show this, Eq. (15) is multiplied by a D and Eq. (4 ) is introducedinto the result producing the expression

Introducing Eq. (9a) into Eq. (16) results in the expression

which, considering Eq. (9b), is an identity. Equation (14) can then beexpressed as

or considering Eq. (13) we can introduce the following notation

If the kinetic and potential energies are expressed in matrix formin terms of angular velocity vectors and rotational transformationmatrices or T(w,c) and V(C), then Lagrange's Equations forquasicoordinates can be expressed as

The operations implicit in the last two terms on the left side of Eq.(20) are actually straightforward according to the identity given byEqs. (13) and (19). Hence, Eq. ( 20 ) permits a straightforward andefficient method of formulating the equations of motion of complexstructures.


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4. ExampleAs an example of the application of Lagrange's equations for

quasicoordinates, consider a flexible spacecraft orbiting the earth.The spacecraft of Fig. 1 consists of the shuttle, which will be assumedto be relatively rigid, and' a flexible appendage extending from theshuttle cargo bay. Considering Fig. 1 and denoting the position of theorigin 0 of the body-fixed reference frame x y z by the vector R, the0 0 0 -position of a point S on the shuttle relative to 0 by r, and the-position of a point A on the appendage relative to 0 by a. Moreover,-the elastic displacement vector of point A is defined as u. The-position of S and A relative to the inertial frame XYZ is R = R+r and-s - -R = R+a+u, respectively. The velocities of points S and A on the-A - - -spacecraft are kS = fi+wxr- - and i = k+wx(a+u)+;, respectively, where fi is-A - - - -the translational velocity and w is the angular velocity of the frame-x y z with respect to the inertial frame. Hence, the kinetic energy of0 0 0the spacecraft is

In order to discretize the system in space, we express the elasticdisplacement vector in the form

where O is a matrix of space-dependent admissible functions and q is a-vector of time-dependent generalized coordinates. Introducing Eq. (22)into Eq. (211, the kinetic energy can be expressed in the matrix form


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T-T -T-T 1 T-+ g 4 + oTj a o w q - g *(w)g iTcA(?)q- I A- "Awhere

Also, m is the mass and I is the mass moment of inertia matrix aboutpoint 0 of the spacecraft. The matrix MA is the mass matrix of theappendage. The matrix C represents a rotational transformation from theinertial frame to the body-fixed frame.

Assuming the origin of the inertial coordinate system coincideswith the center of the gravitational field, the gravitational potentialenergy can be expressed as

where me is the mass of the earth and G is the gravitational constant.The strain energy can be expressed as an energy inner product denoted by[, I as in Meirovitch (1980). The total potential energy then becomes

(26)

where [u,ul includes the potential energy due to centrifugal andgravitational stiffening effects.


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Recognizing that the magnitude of R is large in comparison with the-magnitude of the other vectors in Eq. ( 2 5 ) and ignoring higher orderterms, a binomial expansion permits us to write

Introducing Eq. (27) into Eq. (26) and considering Eq. (221, thepotential energy can be written in the matrix form

where

IRI-

are the stiffness matrix of the appendage, a unit vector in thedirection of R and a matrix of inertia integrals, respectively. Becausethe control forces are most conveniently expressed in the body-fixedframe, the transformation matrix C must be employed in expressing thevirtual work as follows:

where F, M and Q are generalized force vectors in terms of components- ... -


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about xo, yo and zo.Lagrange's equations which govern the translation and elastic

vibration of the spacecraft can be written in the symbolic form

whereas Eq. (20) governs the rotational motion of the structure. Theequations of motion are expressed in detail by Meirovitch and Quinn(1987). The purpose of this example is to demonstrate the use of Eqs.(20) and (19) in producing the equations which govern the rotationalmotion of the structure.

For convenience we shall consider the ten terms of Eq. (23)describing the kinetic energy of the structure separately, or

where the individual Tlterms are defined by the order of Eq. (23).Considering Eq. (20). the terms of Lagrange's equations involvingderivatives with respect to angular velocity can be expressed asfollows:


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where

-J = J(U) = J'.5 [ZUI)r (3 4 ~A

Next, considering Eqs. (13) and (191, the terms of the kinetic andpotential energies, as expressed by Eqs. (231, ( 2 8 ) and (321, involvingderivatives with respect to angular orientation can be expressed asfollows:

"

where the differential operator is defined as

The equations governing the rotational motion of the spacecraft can befound by summing Eqs. (33) and (35d) and subtracting Eqs. (3Sa-c).These equations can be simplified with the introduction of the followingidentity involving skew symmetric matrices:

where a and b are arbitrary vectors and the second tilde over the symbol" -(b) denotes a skew operation on the vector [Gal. Lagrange's equations"governing the rotational motion of the spacecraft can then be expressed


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Gmb

+ (35 J(w)];I- +- hi]151' -Equation (37) is also helpful in simplifying the equations of motiongoverning translation and elastic vibration which have been presented insimplified form by Meirovitch and Quinn (19871.

5. ConclusionsA form of Lagrange's Equations for quasicoordinates (Boltzmann/Hamel

Equations) has been presented which provides a straightforward method offormulating the equations of motion of structures when the energyexpressions are explicit functions of angular orientation. A n identity(Eq. 19) has been introduced which may be utilized if the energies areexpressed in matrix form as functions of angular velocities andcoordinate transformation matrices. This method applies to a largeclass of problems in the dynamics of structures including spacecraft,robotics, ground vehicles and aircraft. The formulation of theequations of motion of a maneuvering flexible spacecraft was shown to berelatively straightforward using this method. A second simplifyingidentity (Eq. 37) was introduced which permits the recognition andcancellation of some like terms which appear in the Lagrangianformulation. The formulation and method, including the simplifyingidentity are suitable for symbolic computation. This permits thedynamic analysis of complex systems.


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REFERENCESHuston, R.L. and Passerello, C.E., "Hultibody StructuralDynamics Including Translation Between the Bodies", Com~ ute rs ndStructures, Vol. 12, Nov. 1980, pp. 713-720.Likins, P.W., "Analytical Dynamics and Nonrigid SpacecraftSimulation," Technical Report 32-1593, July 15, 1974, JetPropulsion Lab, Pasadena, CA.

Likins, P.W., "Quasicoordinate Equations for FlexibleSpacecraft", A I M Journal, Vol. 13, No. 4, April 1975.Meirovitch, L., Methods of Analvtical Dvnamics, McGraw Hill,New York, 1970, pp. 157-162.Meirovitch, L., Com~utational ethods Structural Dvnamics,Sijthoff & Noordhoff, The Netherlands, 1980, pp. 242-252.Meirovitch, L. and Q u i m , R.D., "Equations of Motion forManeuvering Flexible Spacecraft," Journal of Guidance Control,Vol. 10, No. 5, Sept. - Oct. 1987, pp. 453-465.Neimark, Ju. I.and Fufaev, N. A., Dvnamic~of NonholonomicSvstems, American Mathematical Society, Providence, Rhode Island,1972 (Translated from Russian, 1967).Oz, H., Meirovitch, L. and Montgomery, R.C., "On ManeuveringLarge Flexible Spacecraft Using an Annular Momentum ControlDevice", A I M and AAS, Astrodynamics Conference, Danvers Mass.,Aug. 11-13, 1980.Passeron, L., Garnier, Ch. and Sevemec, B., "Dynamic Modelingand Optimal Control Design for Large Flexible Space Structures,"Mechanical Qualification of Larne Flexible Swcecraft Structures,AGARD, Paris, France, 1986, pp. 3-1 to 3-13.Whittaker, E.T., Analvtical Dvnamics of Particles and RinidBodies, Dover Publications, New York, 1944, pp. 41-44.


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EOM for Structures in Terms of Quasi Coordinates

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